Page N TIRUMALA III. Answer any Five questions 5x7=35M 18. Show the points (3,-2), (7,6), (-1,2), (-5, -6) taken in order form a rhombus. 19. Find the points which divide the line segment joining (8, 12 and (12, 8) into four equal parts 20. Find the harmonic conjugate of (7,5) with respect to ((4,2), (9,7) 21. Find the incentre of the triangle formed by (7,9), (3, -7), (-3, 3) 22. The centroid of AABC is (2, 7) the points B, C lie on x,y axes respectively. Then fin a third. Vertk 23. If the Area of the triangle formed by (10,2), (-3,-4), (x, 1) is 5 sq. unit Then finix 24. 2. Simply the expression log 4 2 log4 and compute its value for x = 1/2 4 X 4x. find the points which divide the line segment joining (8,12),(12,8) into four equal parts
Answers
Answer:
consider the problem
If four points are concyclic then 4
th
point must satisfy equation of circle passing through other three points.
Let, x
2
+y
2
+2gx+2fy+c=0 be equation of circle
(5,6),(6,4),and(7,1) must satisfy circle
(5,5)→25+25+10g+10f+c=0
⇒10g+10f+c=−50(i)
(6,4)→36+16+12g+8f+c=0
⇒12g+8f+c=−52(ii)
(7,1)→49+1+14g+2f+c=0
⇒14g+2f+c=−50(iii)
And (ii)−(i)
⇒2g−2f=−2…(iv)
(iii)−(ii)
⇒2g−6f=2…(v)
(iv)−(v)
2g−2f−2g+6f=−2−2
⇒4f=−4
⇒f=−1
Put in (iv)
2g+2=−2
⇒g=−2
10g+10f+c=−50
Put in (i)
−20−10+c=−50
⇒c=−20
Therefore, equation is x
2
+y
2
−4x−2y−20=0
Now, put (−2,4) in R.H.S
4+16+8−8−20=28−28=0
Therefore, (−2,4) satisfies circle passing through (5,5),(6,4)and(7,1)
Hence, (5,5),(6,4),(−2,4)and(7,1) are concyclic
Center is (2,1)
And Radius is
=
4+1−(−20)
=
25
=5
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